In this article we consider the Gauss Legendre Quadrature method for numerical integration over the standard tetrahedron: {(x, y, z)|0 ≤ x, y, z ≤ 1, x + y + z ≤ 1} in the Cartesian three‐dimensional (x, y, z) space. The mathematical transformation from the (x, y, z) space to (ξ, η, ζ) space is described to map the standard tetrahedron in (x, y, z) space to a standard 2‐cube: {(ξ, η, ζ)| − 1 ≤ ζ, η, ζ ≤ 1} in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight coefficients and sampling points. The effectiveness of the formulas is demonstrated by applying them to the integration of three nonpolynomial, three polynomial functions and to the evaluation of integrals for element stiffness matrices in linear three‐dimensional elasticity. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

This paper first presents a Gauss Legendre quadrature method for numerical integration of I ¼ R R T f ðx; yÞ dx dy, where f(x, y) is an analytic function in x, y and T is the standard triangular surface: {(x, y)j0 6 x, y 6 1, x + y 6 1} in the Cartesian two dimensional (x, y) space. We then use a transformation x = x(n, g), y = y(n, g) to change the integral I to an equivalent integral R R S f ðxðn; gÞ; yðn; gÞÞ oðx; yÞ oðn;gÞ dn dg, where S is now the 2-square in (n, g) space: {(n, g)j À 1 6 n, g 6 1}. We then apply the one dimensional Gauss Legendre quadrature rules in n and g variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. We then propose the discretisation of the standard triangular surface T into n 2 right isosceles triangular surfaces T i (i = 1(1)n 2 ) each of which has an area equal to 1/(2n 2 ) units. We have again shown that the use of affine transformation over each T i and the use of linearity property of integrals lead to the result:where H ðX ; Y Þ ¼ P nÂn i¼1 f ðx i ðX ; Y Þ; y i ðX ; Y ÞÞ and x = x i (X, Y) and y = y i (X, Y) refer to affine transformations which map each T i in (x, y) space into a standard triangular surface T in (X, Y) space. We can now apply Gauss Legendre quadrature formulas which are derived earlier for I to evaluate the integral I ¼ 1 n 2 RR T H ðX ; Y Þ dX dY . We observe that the above procedure which clearly amounts to Composite Numerical Integration over T and it converges to the exact value of the integral RR T f ðx; yÞ dx dy, for sufficiently large value of n, even for the lower order Gauss Legendre quadrature rules. We have demonstrated this aspect by applying the above explained Composite Numerical Integration method to some typical integrals.

This paper is concerned with curved boundary triangular elements having one curved side and two straight sides. The curved elements considered here are the 6-node (quadratic), 10-node (cubic), 15-node (quartic) and 21-node (quintic) triangular elements. On using the isoparametric coordinate transformation, these curved triangles in the global (x, y) coordinate system are mapped into a standard triangle: {( , )/0 , 1, + 1} in the local coordinate system ( , ). Under this transformation curved boundary of these triangular elements is implicitly replaced by quadratic, cubic, quartic and quintic arcs. The equations of these arcs involve parameters, which are the coordinates of points on the curved side. This paper deduces relations for choosing the parameters in quartic and quintic arcs in such a way that each arc is always a parabola which passes through four points of the original curve, thus ensuring a good approximation. The point transformations which are thus determined with the above choice of parameters on the curved boundary and also in turn the other parameters in the interior of curved triangles will serve as a powerful subparametric coordinate transformation for higher order curved triangular elements with one curved side and two straight sides.

This paper presents a Gaussian Quadrature method for the evaluation of the triple integral R R T R f ðx; y; zÞ dx dy dz, where f ðx; y; zÞ is an analytic function in x, y, z and T refers to the standard tetrahedral region: fðx; y; zÞ 0 6 x; y; z 6 j 1; x þ y þ z 6 1g: in three space ðx; y; zÞ. Mathematical transformation from ðx; y; zÞ space to ðU ; V ; W Þ space map the standard tetrahedron T in ðx; y; zÞ space to a standard 1-cube: fðU ; V ; W Þ=0 6 U ; V ; W 6 1g in ðU ; V ; W Þ space. Then we use the product of Gauss Legendre and Gauss Jacobi weight coefficients and abscissas to arrive at an efficient quadrature rule over the standard tetrahedral region T. We have then demonstrated the application of the derived quadrature rules by considering the evaluation of some typical triple integrals over the region T.

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